Russell's theory of descriptions

Russell's
theory of descriptions is a theory about how to understand definite
descriptions: that is, descriptions that are particularly suited to
referring to single things, such as:

the cow,

the president of the USA,

the nicest cow in the world,

7­2

John's best friend.

Russell's
proposal is that we understand "The cow is brown" as saying
that there is exactly one cow, and it is brown. (Or, which comes to the
same thing, there is exactly one cow, and all cows are brown.). Thus:

$x[[CxÙ"y[Cy®y=x]]ÙBx].

As it stands
this may strike one as obviously wrong: surely someone who says that the
cow is brown, does not mean that there is exactly one cow in the whole
wide world, and that it is brown. However, we may easily suppose that
the domain in question is implicitly a good deal more limited than everything
there is; plausibly it will be restricted (rather vaguely, perhaps), to
things in the vicinity, or some such thing. In fact the context will help
to determine the domain; it may be things in the picture, or, things on
Farmer Giles's farm (if we happen to be talking about that), or…

One might
also object to the proposal on the grounds that someone who says that
the cow is brown will not actually be saying that there is exactly
one cow etc. But, by way of defence, one might say that at any rate, if
there is not exactly one cow, what the person says will not be true.
So, supposing that the sentence is either true of false, the truth-value
of the proposed paraphrase will always agree with that of the original;
and that will be enough for our purposes.

Because it
will be useful for further formalisations, consider how we might formalise

Buttercup is the cow.

Using "b"
for "Buttercup", we could say,

$x[[CxÙ"y[Cy®y=x]]Ùx=b],

(i.e. "The
cow is Buttercup".)

More compactly
we can say:

[CbÙ"y[Cy®y=b]].

So now, if
we want to formalise

The farmer bought the cow,

We can say,
as a first step,

$x[x is the farmer Ù$y[y is the cow Ù
x bought y]].

(Notice that
we have to make sure that "x bought y" is in the scope of both
the quantifiers.) Or, if you prefer:

$x$y[x is the farmer Ù
[y is the cow Ù x bought y]].

And we now
replace "x is the farmer" and "y is the cow" with
expressions which follow the pattern of the formalisation of "Buttercup
is the cow". Thus: